Page 106 - MATINF Nr. 13-14

106 M.N. Popescu

8. c. Media variabilei aleatoare Z n este

n n n−k l n n−k l
X X 1 X (−1) X 1 X (−1)
M (Z n ) = k · p k = k · · = ·
k! l! (k − 1) ! l !
k=0 k=0 l=0 k=1 l=0
ñ 0 1 2 n−2 n−1 ô
1 (−1) (−1) (−2) (−1) (−1)
= · + + + . . . + +
0! 0! 1! 2! (n − 2)! (n − 1)!
ñ 0 1 2 n−2 ô
1 (−1) (−1) (−2) (−1)
+ · + + + . . . +
1! 0! 1! 2! (n − 2)!
. . .

1 (−1) 0
+ · ,
(n − 1)! 0!

a
deci regrup˘am dup˘ diagonale secundare
1 (−1) 0
M (Z n ) = ·
0! 0!
ñ 0 1 ô
1 (−1) 1 (−1)
+ · + ·
1! 0! 0! 1!
ñ 0 1 2 ô
1 (−1) 1 (−1) 1 (−1)
+ · + · + ·
2! 0! 1! 1! 0! 2!
. . .

ñ 0 1 n−1 ô
1 (−1) 1 (−1) 1 (−1)
+ · + · + . . . + ·
(n − 1)! 0! (n − 2)! 1! 0! (n − 1)!
s , i am obt , inut suma part , ial˘ de ordinul (n − 1) a seriei produs
a
! !
n
X 1 X (−1)
· .
n! n!
n≥0 n≥0
Prin urmare
! !
∞ ∞ n
X 1 X (−1)
lim M (Z n ) = · = e · e −1 = 1.
n→∞ n! n!
n=0 n=0
9.

Å ã
1
• Pentru n = 1, S 1 cont , ine doar permutarea σ = care se reduce la un punct fix, deci
1
ω (σ) = 1. Prin urmare
1 X 1
ω (σ) = · 1 = 1.
1! 1!
σ∈S 1
a
• Pentru n = 2, S 2 cont , ine doar dou˘ permut˘ari:

Å ã
1 2
σ 1 = care are dou˘ puncte fixe, deci ω (σ 1 ) = 2,
a
1 2